How this calculation works
A service team can have average capacity above average demand and still make some arrivals wait. Random arrival and service timing create a queue. Erlang C describes that waiting behavior under a specific steady-state multi-server model. This page reports waiting probability, service within an entered waiting-time target, mean queue wait and utilization for a chosen agent count. It does not automatically select a staffing recommendation.
Offered load = arrivals × handle time/3600; Erlang C yields wait probability; P(wait≤t)=1−C exp[−(agents×service rate−arrivals)t]A worked example
Thirty arrivals per hour with sixty-second mean service and one available agent give offered load 0.5 and utilization fifty percent. In the M/M/1 special case, waiting probability is 0.5. Immediate-service probability is therefore fifty percent. Mean queue waiting time including nonwaiting arrivals is sixty seconds, and the mean number queued is 0.5.
Good to know
Poisson arrivals, independent exponential service times, FIFO, infinite waiting room and no abandonment are assumed. Skill routing, shrinkage, blocked calls and time-varying bursts are absent. Real staffing needs breaks, schedule constraints and empirical validation. A modeled service percentage is not a guarantee that an actual team will achieve the same outcome.
Reference: NPTEL — M/M/m queue and Erlang C (opens in a new tab) ↗
What this calculation answers
A service team can have average capacity above average demand and still make some arrivals wait. Random arrival and service timing create a queue. Erlang C describes that waiting behavior under a specific steady-state multi-server model.
This page reports waiting probability, service within an entered waiting-time target, mean queue wait and utilization for a chosen agent count. It does not automatically select a staffing recommendation.
The Production Little law WIP cycle time and throughput calculator links stable average queue length, flow and time. Erlang C supplies additional stochastic waiting assumptions that the average identity alone does not contain.
Use the right input conventions
Arrivals are per hour and mean handling time is in seconds. Agents are identical and continuously available for service. Offered load must be strictly below agent count for a stable finite-wait result.
The target is time waiting before service begins, not total time including handling. At a zero-second target, service level is the probability of being served immediately.
Interpret the result
Mean queue wait includes arrivals that wait zero time. It is therefore different from average wait conditional on having to queue. Little’s law connects that all-arrival mean to average queue length.
The implementation uses a stable Erlang B recursion to obtain Erlang C instead of large factorials. This numerical choice does not relax the underlying queueing assumptions.
Work through the example
Thirty arrivals per hour with sixty-second mean service and one available agent give offered load 0.5 and utilization fifty percent. In the M/M/1 special case, waiting probability is 0.5.
Immediate-service probability is therefore fifty percent. Mean queue waiting time including nonwaiting arrivals is sixty seconds, and the mean number queued is 0.5.
Where the model stops
Poisson arrivals, independent exponential service times, FIFO, infinite waiting room and no abandonment are assumed. Skill routing, shrinkage, blocked calls and time-varying bursts are absent.
Real staffing needs breaks, schedule constraints and empirical validation. A modeled service percentage is not a guarantee that an actual team will achieve the same outcome.
A few questions, answered
Does the mean queue wait include immediately served arrivals?
Yes. It averages over all arrivals, including those with zero waiting time.
What changes in the comparison?
Add a second continuously available agent and a twenty-second waiting target.